Abstract
If \( \mathbb{T} \) has a right-scattered minimum m, define \( \mathbb{T}_\kappa : = \mathbb{T} - \{ m\} \) ; otherwise, set \( \mathbb{T}_\kappa = \mathbb{T} \) . The backwards graininess \( \nu :\mathbb{T}_\kappa \to \mathbb{R}_0^ + \) is defined by
For \( f:\mathbb{T} \to \mathbb{R} \) and \( t \in \mathbb{T}_\kappa \) , define the nabla derivative [42] of f at t, denoted f ∇(t), to be the number (provided it exists) with the property that given any ε > 0, there is a neighborhood U of t such that
for all s € U. For \( \mathbb{T} = \mathbb{R} \) , we have f ∇=f′, the usual derivative, and for \( \mathbb{T} = \mathbb{Z} \) we have the backward difference operator, f ∇(t)=∇f(t):=f(t)-f(t-1). Note that the nabla derivative is the alpha derivative when α = p. Many of the results in this chapter can be generalized to the alpha derivative case. Many of the results in this chapter can be found in [35, 37].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Anderson, D., Bullock, J., Erbe, L., Peterson, A., Tran, H. (2003). Nabla Dynamic Equations. In: Bohner, M., Peterson, A. (eds) Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8230-9_3
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8230-9_3
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6502-3
Online ISBN: 978-0-8176-8230-9
eBook Packages: Springer Book Archive